The Uniform Integrability of Martingales. On a Question by Alexander Cherny

TL;DR

This paper investigates conditions under which a stochastic process is a uniformly integrable martingale, providing counterexamples and establishing new integrability criteria involving randomized stopping times.

Contribution

It demonstrates that nonnegativity is crucial for uniform integrability and introduces an integrability condition that ensures the implication holds with randomized stopping times.

Findings

Counterexample showing nonnegativity is essential

New integrability condition involving limit inferior of |X|

Randomized stopping times satisfy the integrability assumption

Abstract

Let $X$ be a progressively measurable, almost surely right-continuous stochastic process such that $X_\tau \in L^1$ and $E[X_\tau] = E[X_0]$ for each finite stopping time $\tau$. In 2006, Cherny showed that $X$ is then a uniformly integrable martingale provided that $X$ is additionally nonnegative. Cherny then posed the question whether this implication also holds even if $X$ is not necessarily nonnegative. We provide an example that illustrates that this implication is wrong, in general. If, however, an additional integrability assumption is made on the limit inferior of $|X|$ then the implication holds. Finally, we argue that this integrability assumption holds if the stopping times are allowed to be randomized in a suitable sense.